Abstracts

Week of March 23, 2025

March 24, 2025
Group Actions and Dynamics Moments of Margulis functions and values of ternary quadratic forms 4:00pm -
KT207

 The Oppenheim conjecture, proved by Margulis in 1986, states that for a non-degenerate indefinite irrational quadratic form Q in $n \geq 3$ variables, the image set $Q(Z^n)$  of integral vectors is a dense subset of the real line. Determining the distribution of values of an indefinite quadratic form at integral points asymptotically is referred to as quantitative Oppenheim conjecture. The quantitative Oppenheim conjecture was established by Eskin, Margulis, and Mozes for quadratic forms in $n \geq 4$ variables. In this talk, we discuss the quantitative Oppenheim conjecture for ternary quadratic forms (n=3). The main ingredient of the proof is a uniform boundedness result for the moments of Margulis functions over expanding translates of a unipotent orbit in the space of 3-dimensional lattices, under suitable Diophantine conditions of the initial unipotent orbit.
Geometry, Symmetry and Physics Chiral Differential Operators on the Basic Affine Space 4:30pm -
KT 801

Although vertex algebras originated in 2D conformal field theory, recent developments show that they also arise in 3D and 4D quantum field theories. Such vertex algebras often appear as chiralizations of symplectic singularities.

To construct new examples of these chiralizations, we study chiral differential operators on the basic affine space G/U, where G is a simple, simply connected algebraic group of type ADE, and U is a maximal unipotent subgroup of G. We show that the associated variety of these vertex algebras is isomorphic to the affine closure of G/U, which is a symplectic singularity, as conjectured by Ginzburg and Kazhdan and later proved by Jia and Gannon.

This is a joint work in progress with Xuanzhong Dai and Bailin Song.
March 25, 2025
Geometry & Topology Truncated braid groups 4:00pm -
KT 207

In the 1950s, Coxeter considered the quotients of braid groups given by adding the relation that all half Dehn twist generators have some fixed, finite order. He found a remarkable formula for the order of these groups in terms of some related Platonic solids. Despite the inspiring apparent connection between these objects, Coxeter's proof boils down to a finite case check that reveals nothing about the structure present. I'll explain recent work that gives an interpretation of the truncated 3-strand braid group that makes the connection with Platonic solids clear, using down-to-earth geometric and algebraic topological tools. This is joint work with Tahsin Saffat.
Geometry, Symmetry and Physics Quantum Category O and Categorification of Periodic Hecke Module, Lecture 1 4:00pm to 6:00pm -
KT 801

The BGG category O plays an important role in the study of representations of semisimple Lie algebras. Its connection to the Hecke category is a starting point of Geometric Representation Theory. In this lecture, I will introduce a version of category O for quantum groups at roots of unity. I will explain a derived equivalence from (the principal block of) quantum category O to the affine Hecke category. Under the equivalence, the highest weight structure of quantum category O provides a categorification of the “periodic Hecke module”.

In the first lecture, I will briefly review the classical story for BGG category O. Then I will give an introduction to the quantum category O with motivations.
March 26, 2025
Colloquium Teaching dynamics to biology undergraduates 4:00pm -
KT 101

There is an urgent need to reform how we introduce math to beginning students in Life Sciences. The usual “Calculus for Life Sciences”, which is a watered down version of Calculus I, possibly including some trivial biological examples, has failed to inspire students. Even worse, the math gateway courses into the life sciences serve as powerful filters keeping women and underrepresented minorities out of the life sciences and medicine. Recently, there have been calls, from all the leading voices in US biology and medicine, for a new approach to mathematics for biology.

We designed such a course, and are currently teaching it to ~2500 students/year at UCLA, which introduces students, on day 1, to the concept of modeling a system that has multiple interacting variables and nonlinear relations. The student quickly learns that models give rise to ‘change equations’, that tell you, at any point in state space, where to head and how fast. Throughout, there is a strong emphasis on biological applications of these concepts, such as bistability (“biological switches”), feedback behaviors in physiology and ecology, qualitative changes in system behavior (i.e. bifurcations) and oscillations in, for example, insulin and glucose levels and in biological populations.

March 27, 2025
Analysis Optimal stability for geometric inequalities 4:00pm -
KT 207

In this talk, we will discuss some recent works on stability for some sharp geometric inequalities, including optimal lower bounds for higher and fractional order Sobolev inequality and Hardy-Littlewood-Sobolev inequality in Euclidean space and log-Sobolev inequality on the sphere. Stability for Caffarelli-Kohn-Nirenberg inequality will also be discussed if time permits.
Quantum Topology and Field Theory A quantization of the complex Chern-Simons invariant 4:30pm -
KT 801

This talk is about a new family of geometric quantum link invariants that depend on both a link in S^3 and a flat sl_2 connection on its complement. When the connection is trivial they recover the Kashaev invariant (a certain evaluation of the colored Jones polynomial). More generally they can be understood as a quantization of the complex Chern-SImons invariant of the flat connection (aka complex volume), whose real and imaginary parts are the volume and Chern-Simons invariant of the hyperbolic structure determined by the connection. In this talk I will discuss the construction of these invariants using the representation theory of quantum sl_2 and how the classical complex Chern-Simons invariant arises naturally in this context, then sketch some proposed connections with quantum SL_2(C) Chern-Simons theory. Given time I will also discuss connections with the Volume Conjecture. This talk is based on joint work with Nicolai Reshetikhin.
March 28, 2025
Friday Morning Seminar How you think on a function defined on 0,1,…,N-1? 10:00am -
KT 801

Between thousand to million times per day, your cellphone calculates the Fourier Transform (FT) of certain functions defined on 0,1,…,N-1, with N large (order of magnitude of thousands and more). The calculation is done using the Fast Fourier Transform (FFT) - discovered by Cooley–Tukey in 1965 and by Gauss in 1805. In the lecture I want to advertise a beautiful way—due to Auslander-Tolimieri—to obtain the FFT as a natural consequence of an answer to the following:

Question: How to think on the space of functions on the set 0,1,…,N-1? 

Engineers tell us that there are two answers for this question: 

(A) as functions on that set, where 0,1,…,N-1 regarded as times; 

and,

(B) as functions on that set, where 0,1,…,N-1 regarded frequencies; 

and then the FT is an operator translating between the two spaces. In the lecture, I will explain that there is another answer, i.e., a not so well-known third space (C), of arithmetic nature, that also gives an answer to the above question, and then the FFT appears simply as the composition of two operators: the one translating between spaces (A) and (C), and the one that translates (C) to (B).  

Remark: The lecture is prepared to be understood to anyone who is familiar with basic linear algebra. In particular, advanced undergraduate students, from computer science, engineering, mathematics, physics, etc, are more than welcome to attend.
Algebra and Geometry lecture series Fourier-Mukai Transformations of certain upward flows 3:00pm -
KT801

Recently there was a paper of Hausel and Hitchin which developed the theory of upward flows/attracting loci in the moduli space of semistable Higgs bundles, and used them to for example to compute multiplicities of components of the global nilpotent cone. They also gave an incomplete picture of what the mirror of certain upward flows look like under some version of homological mirror symmetry. We give a more complete explanation of this mirror, using the duality theories for compactified Jacobians developed by Arinkin, Melo-Rapagnetta-Viviani, and Maulik-Shen-Yin.